Why are keyframes in global coords, instead of local coords? (Blender 2.70a)
Sometimes this problem can be "solved" by choosing a different Rotation Mode.  "XYZ Euler" is popular (because it is the default), but sometimes people pick a different one, either to avoid gimbal lock, or to achieve a particular animation effect.

However, the general purpose solution is to use parenting, either the genuine object parent field, or a Child Of constraint.  You tilt the "parent" such that one of its axes is what you want to rotate around (lined up with the spindle of the record player), then animate the rotation of the record around a single axis.  As you can see from the many comments, there are a couple of different ways to accomplish the effect you want, and each one has its pros and cons.

The Humane Rigging series has a few videos that might expand your understanding, the one I'd start with is Humane Rigging Chapter 3 part 3 although you will probably want to watch part 2 and 4 when you have some spare time.